This paper deals with the study of the existence of nontrivial solutions and multiple solutions of the semilinear elliptic boundary value problem \[ \begin{cases} -\Delta u=\lambda_k u+g(u) \quad & \text{in } \Omega,\\ u=0\quad & \text{on }\partial \Omega, \end{cases} \tag{1} \] where \(\Omega\subset \mathbb{R}^d\) is a bounded open domain with smooth boundary and \(g\in C^1(\mathbb{R}^1, \mathbb{R}^1)\) satisfies \(g(0)=0\). Under some suitable assumptions on \(g\) and \(f'(0)<\lambda_1\), where \(f(t):= \lambda_k t+g(t)\), the author shows that (1) has at least three nontrivial solutions in which one is positive and one is negative. To this end the author uses Morse theory and critical groups with the minimax methods such as mountain pass lemma and the local linking.